I have a reference image A with a known position and I want to calculate the relative position of the camera at image B (i.e. tx, ty, tz in meters). The images are taken with the same camera so the camera matrix stays the same. I'm using SIFT to detect and compute the keypoints and descriptors in both images and match them with FLANN. From there I can get the homography matrix which I decompose with cv::decomposeHomography(..). This function is based on this paper: PDF.
In this paper it is stated, that the translation matrix is normalized by d*, which is the plane depth.
In order to get the correct translation I need to know the plane depth. Is there a way to get this without knowing the size of an object found in the image?
The 3D translation computed using homography decomposition is only computable up to an unknown scale factor. This is a classical problem with computing 3D geometry from monocular images using only apparent motion in the images. Typically 3D reconstructions from monocular images are called metric reconstructions for this reason (rather than Euclidean reconstructions where scale is resolved). To resolve the scale factor some more information is needed, such as knowing the depth of a point on the plane or the distance moved by the camera between images.
Related
I am interested in finding the Rotation Matrix of an Aruco Marker from a Stereo Camera.
I know that estimateposesinglemarkers gives a Rotation Vector (which can be converted to matrix via Rodrigues)and Translation Vector but the values are not that stable and is supposedly written for MonoCam.
I can get Stable 3D points of the Marker from a Stereo Camera, however i am struggling in creating a Rotation Matrix. My Main goal is to achieve what Ali has achieved in this following blog Relative Position of Aruco Markers.
I have tried working with Euler Angles from here by creating a plane of the Aruco Marker from the 3D points that i get from the Stereo Camera but in vain.
I know my algorithm is failing because the values of the Relative Co-ordinates keeps on changing on moving the camera which should not happen as the Relative Co-ordinates b/w the Markers Should remain Constant.
I have a properly Calibrated camera with all the required matrices.
I tried using SolvePnP, but i believe it gives Rvecs and Tvecs which when combined together brings points from the model coordinate system to the camera coordinate system.
Any idea on how i can create the Rotation Matrix of the Marker with my fairly Stable 3D points so that on moving the camera, the relative Co-ordinates doesn't Change ?
Thanks in Advance.
I am willing to perform a 360° Panorama stitching for 6 fish-eye cameras.
In order to find the relation among cameras I need to compute the Homography Matrix. The latter is usually computed by finding features in the images and matching them.
However, for my camera setup I already know:
The intrinsic camera matrix K, which I computed through camera calibration.
Extrinsic camera parameters R and t. The camera orientation is fixed and does not change at any point. The cameras are located on a circle of known diameter d, being each camera positioned with a shift of 60° degrees with respect to the circle.
Therefore, I think I could manually compute the Homography Matrix, which I am assuming would result in a more accurate approach than performing feature matching.
In the literature I found the following formula to compute the homography Matrix which relates image 2 to image 1:
H_2_1 = (K_2) * (R_2)^-1 * R_1 * K_1
This formula only takes into account a rotation angle among the cameras but not the translation vector that exists in my case.
How could I plug the translation t of each camera in the computation of H?
I have already tried to compute H without considering the translation, but as d>1 meter, the images are not accurate aligned in the panorama picture.
EDIT:
Based on Francesco's answer below, I got the following questions:
After calibrating the fisheye lenses, I got a matrix K with focal length f=620 for an image of size 1024 x 768. Is that considered to be a big or small focal length?
My cameras are located on a circle with a diameter of 1 meter. The explanation below makes it clear for me, that due to this "big" translation among the cameras, I have remarkable ghosting effects with objects that are relative close to them. Therefore, if the Homography model cannot fully represent the position of the cameras, is it possible to use another model like Fundamental/Essential Matrix for image stitching?
You cannot "plug" the translation in: its presence along with a nontrivial rotation mathematically implies that the relationship between images is not a homography.
However, if the imaged scene is and appears "far enough" from the camera, i.e. if the translations between cameras are small compared to the distances of the scene objects from the cameras, and the cameras' focal lengths are small enough, then you may use the homography induced by a pure rotation as an approximation.
Your equation is wrong. The correct formula is obtained as follows:
Take a pixel in camera 1: p_1 = (x, y, 1) in homogeneous coordinates
Back project it into a ray in 3D space: P_1 = inv(K_1) * p_1
Decompose the ray in the coordinates of camera 2: P_2 = R_2_1 * P1
Project the ray into a pixel in camera 2: p_2 = K_2 * P_2
Put the equations together: p_2 = [K_2 * R_2_1 * inv(K_1)] * p_1
The product H = K2 * R_2_1 * inv(K1) is the homography induced by the pure rotation R_2_1. The rotation transforms points into frame 2 from frame 1. It is represented by a 3x3 matrix whose columns are the components of the x, y, z axes of frame 1 decomposed in frame 2. If your setup gives you the rotations of all the cameras with respect to a common frame 0, i.e. as R_i_0, then it is R_2_1 = R_2_0 * R_1_0.transposed.
Generally speaking, you should use the above homography as an initial estimation, to be refined by matching points and optimizing. This is because (a) the homography model itself is only an approximation (since it ignores the translation), and (b) the rotations given by the mechanical setup (even a calibrated one) are affected by errors. Using matched pixels to optimize the transformation will minimize the errors where it matters, on the image, rather than in an abstract rotation space.
i am completing my thesis related opencv.
I want to measure real size of object (mm) with single camera but i have problem with convert the camera's natural units (pixels) and the real world units!!!
After calibrate camera, i have:
Camera matrix (3x3)
Distortion coefficients
Extrinsic parameters [rotation vector(1x3) + translation vector(1x3)]
I have read following link but i can't find out formula to convert unit.
https://docs.opencv.org/2.4/modules/calib3d/doc/camera_calibration_and_3d_reconstruction.html
Example about measure size of object
Any sugguestion???
Thanks so much.
As mentioned in the comments, you need the distance to the object to obtain 3D coordinates from pixels. A possible workflow would be:
Rectify the image using the distortion parameters, i.e., correct the distortion caused by the camera.
Deproject the pixels into 3D points in the camera coordinate frame using the camera matrix. For this you can multiply the inverse of the 3x3 camera matrix with a vector containing the pixels [pixel_x, pixel_y, 1]^T. If you multiply the result [x', y', 1]^T with the depth, i.e., the z-component you obtain the 3D point in the camera coordinate frame.
Transform the point from the camera coordinate frame into the world coordinate frame using the extrinsics parameters.
Obtaining the depth values from an image alone is not possible. The only option is to use some additional information. Maybe your object is placed on a table and you know the distance between the camera and the table.
To measure distances between the camera and a table or even the object itself you could use Aruco markers, which are also available within openCV.
I have a single calibrated camera pointing at a checkerboard at different locations with Known
Camera Intrinsics. fx,fy,cx,cy
Distortion Co-efficients K1,K2,K3,T1,T2,etc..
Camera Rotation & Translation (R,T) from IMU
After Undistortion, I have computed Point correspondence of checkerboard points in all the images with a known camera-to-camera Rotation and Translation vectors.
How can I estimate the 3D points of the checkerboard in all the images?
I think OpenCV has a function to do this but I'm not able to understand how to use this!
1) cv::sfm::triangulatePoints
2) triangulatePoints
How to compute the 3D Points using OpenCV?
Since you already have the matched points form the image you can use findFundamentalMat() to get the fundamental matrix. Keep in mind you need at least 7 matched points to do this. If you have more then 8 points CV_FM_RANSAC might be the best option.
Then use cv::sfm::projectionsFromFundamental() to find the projection matrix for each image, check if the projection matrix is valid (ex.check if the points are in-front of the camera).
then feed the projections and the points it into cv::sfm::triangulatePoints().
Hope this helps :)
Edit
The rotation and translation matrix are needed to change reference frames because the camera moves in SFM. The reference frame is at the position of the camera. Transforms are needed to make sure the position of the points a coherent(under the same reference frame which is usually the reface frame of the camera in the first image), so all the points are in the same coordinate system.
IE. To relate the point gathered by the second frame to the first frame, the third to second frame and so on.
So basically you can use the R and T vector to construct a transform matrix for each frame and multiplying it with your points to put them in the reface frame of the camera in the first frame.
I am working on building 3D point cloud from features matching using OpenCV3.1 and OpenGL.
I have implemented 1) Camera Calibration (Hence I am having Intrinsic Matrix of the camera) 2) Feature extraction( Hence I have 2D points in Pixel Coordinates).
I was going through few websites but generally all have suggested the flow for converting 3D object points to pixel points but I am doing completely backword projection. Here is the ppt that explains it well.
I have implemented film coordinates(u,v) from pixel coordinates(x,y)(With the help of intrisic matrix). Can anyone shed the light on how I can render "Z" of camera coordinate(X,Y,Z) from the film coordinate(x,y).
Please guide me on how I can utilize functions for the desired goal in OpenCV like solvePnP, recoverPose, findFundamentalMat, findEssentialMat.
With single camera and rotating object on fixed rotation platform I would implement something like this:
Each camera has resolution xs,ys and field of view FOV defined by two angles FOVx,FOVy so either check your camera data sheet or measure it. From that and perpendicular distance (z) you can convert any pixel position (x,y) to 3D coordinate relative to camera (x',y',z'). So first convert pixel position to angles:
ax = (x - (xs/2)) * FOVx / xs
ay = (y - (ys/2)) * FOVy / ys
and then compute cartesian position in 3D:
x' = distance * tan(ax)
y' = distance * tan(ay)
z' = distance
That is nice but on common image we do not know the distance. Luckily on such setup if we turn our object than any convex edge will make an maximum ax angle on the sides if crossing the perpendicular plane to camera. So check few frames and if maximal ax detected you can assume its an edge (or convex bump) of object positioned at distance.
If you also know the rotation angle ang of your platform (relative to your camera) Then you can compute the un-rotated position by using rotation formula around y axis (Ay matrix in the link) and known platform center position relative to camera (just subbstraction befor the un-rotation)... As I mention all this is just simple geometry.
In an nutshell:
obtain calibration data
FOVx,FOVy,xs,ys,distance. Some camera datasheets have only FOVx but if the pixels are rectangular you can compute the FOVy from resolution as
FOVx/FOVy = xs/ys
Beware with Multi resolution camera modes the FOV can be different for each resolution !!!
extract the silhouette of your object in the video for each frame
you can subbstract the background image to ease up the detection
obtain platform angle for each frame
so either use IRC data or place known markers on the rotation disc and detect/interpolate...
detect ax maximum
just inspect the x coordinate of the silhouette (for each y line of image separately) and if peak detected add its 3D position to your model. Let assume rotating rectangular box. Some of its frames could look like this:
So inspect one horizontal line on all frames and found the maximal ax. To improve accuracy you can do a close loop regulation loop by turning the platform until peak is found "exactly". Do this for all horizontal lines separately.
btw. if you detect no ax change over few frames that means circular shape with the same radius ... so you can handle each of such frame as ax maximum.
Easy as pie resulting in 3D point cloud. Which you can sort by platform angle to ease up conversion to mesh ... That angle can be also used as texture coordinate ...
But do not forget that you will lose some concave details that are hidden in the silhouette !!!
If this approach is not enough you can use this same setup for stereoscopic 3D reconstruction. Because each rotation behaves as new (known) camera position.
You can't, if all you have is 2D images from that single camera location.
In theory you could use heuristics to infer a Z stacking. But mathematically your problem is under defined and there's literally infinitely many different Z coordinates that would evaluate your constraints. You have to supply some extra information. For example you could move your camera around over several frames (Google "structure from motion") or you could use multiple cameras or use a camera that has a depth sensor and gives you complete XYZ tuples (Kinect or similar).
Update due to comment:
For every pixel in a 2D image there is an infinite number of points that is projected to it. The technical term for that is called a ray. If you have two 2D images of about the same volume of space each image's set of ray (one for each pixel) intersects with the set of rays corresponding to the other image. Which is to say, that if you determine the ray for a pixel in image #1 this maps to a line of pixels covered by that ray in image #2. Selecting a particular pixel along that line in image #2 will give you the XYZ tuple for that point.
Since you're rotating the object by a certain angle θ along a certain axis a between images, you actually have a lot of images to work with. All you have to do is deriving the camera location by an additional transformation (inverse(translate(-a)·rotate(θ)·translate(a)).
Then do the following: Select a image to start with. For the particular pixel you're interested in determine the ray it corresponds to. For that simply assume two Z values for the pixel. 0 and 1 work just fine. Transform them back into the space of your object, then project them into the view space of the next camera you chose to use; the result will be two points in the image plane (possibly outside the limits of the actual image, but that's not a problem). These two points define a line within that second image. Find the pixel along that line that matches the pixel on the first image you selected and project that back into the space as done with the first image. Due to numerical round-off errors you're not going to get a perfect intersection of the rays in 3D space, so find the point where the ray are the closest with each other (this involves solving a quadratic polynomial, which is trivial).
To select which pixel you want to match between images you can use some feature motion tracking algorithm, as used in video compression or similar. The basic idea is, that for every pixel a correlation of its surroundings is performed with the same region in the previous image. Where the correlation peaks is, where it likely was moved from into.
With this pixel tracking in place you can then derive the structure of the object. This is essentially what structure from motion does.